GPU Computing in Julia

Biostat/Biomath M257

Author

Dr. Hua Zhou @ UCLA

Published

April 16, 2024

This lecture introduces GPU computing in Julia.

1 GPGPU

GPUs are ubiquitous in modern computers. Following are NVIDIA GPUs on today’s typical computer systems.

NVIDIA GPUs	H100 PCIe	RTX 6000	RTX 5000
			$RTX 5000$
Computers	servers, cluster	desktop	laptop

Main usage	scientific computing	daily work, gaming	daily work
Memory	80 GB	48 GB	16 GB
Memory bandwidth	2 TB/sec	960 GB/sec	576 GB/sec
Number of cores	???	???	???
Processor clock	??? GHz	??? GHz	??? GHz
Peak DP performance	26 TFLOPS	??? TFLOPS	??? TFLOPS
Peak SP performance	51 TFLOPS	91.1 TFLOPS	42.6 TFLOPS

2 GPU architecture vs CPU architecture

GPUs contain 1000s of processing cores on a single card; several cards can fit in a desktop PC
Each core carries out the same operations in parallel on different input data – single program, multiple data (SPMD) paradigm
Extremely high arithmetic intensity if one can transfer the data onto and results off of the processors quickly

3 GPGPU in Julia

GPU support by Julia is under active development. Check JuliaGPU for currently available packages.

There are multiple paradigms to program GPU in Julia, depending on the specific hardware.

CUDA is an ecosystem exclusively for Nvidia GPUs. There are extensive CUDA libraries for scientific computing: CuBLAS, CuRAND, CuSparse, CuSolve, CuDNN, …

The CUDA.jl package allows defining arrays on Nvidia GPUs and overloads many common operations.
The AMDGPU.jl package allows defining arrays on AMD GPUs and overloads many common operations.
The Metal.jl package allows defining arrays on Apple Silicon and overloads many common operations.
The oneAPI.jl package allows defining arrays on Intel GPUs and overloads many common operations.

I’ll illustrate using Metal.jl on my MacBook Pro running MacOS Sonoma 14.4.1. It has Apple M2 chip with 38 GPU cores.

versioninfo()

Julia Version 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin22.4.0)
  CPU: 12 × Apple M2 Max
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, apple-m1)
Threads: 8 default, 0 interactive, 4 GC (on 8 virtual cores)
Environment:
  JULIA_NUM_THREADS = 8
  JULIA_EDITOR = code

Load packages:

using Pkg

Pkg.activate(pwd())
Pkg.instantiate()
Pkg.status()

  Activating project at `~/Documents/github.com/ucla-biostat-257/2024spring/slides/09-juliagpu`

Status `~/Documents/github.com/ucla-biostat-257/2024spring/slides/09-juliagpu/Project.toml`
  [6e4b80f9] BenchmarkTools v1.5.0
  [bdcacae8] LoopVectorization v0.12.169
  [dde4c033] Metal v1.1.0
  [37e2e46d] LinearAlgebra

4 Query GPU devices in the system

using Metal

Metal.versioninfo()

macOS 14.4.1, Darwin 23.4.0

Toolchain:
- Julia: 1.10.2
- LLVM: 15.0.7

Julia packages: 
- Metal.jl: 1.1.0
- LLVMDowngrader_jll: 0.1.0+1

1 device:
- Apple M2 Max (4.000 GiB allocated)

5 Transfer data between main memory and GPU

using Random
Random.seed!(257)

# generate SP data on CPU
x = rand(Float32, 3, 3)
# transfer data form CPU to GPU
xd = MtlArray(x)

3×3 MtlMatrix{Float32, Private}:
 0.145793  0.939801  0.479926
 0.567772  0.577251  0.81655
 0.800538  0.38893   0.914135

# generate array on GPU directly
# yd = Metal.ones(3, 3)
yd = MtlArray(ones(Float32, 3, 3))

3×3 MtlMatrix{Float32, Private}:
 1.0  1.0  1.0
 1.0  1.0  1.0
 1.0  1.0  1.0

# collect data from GPU to CPU
x = collect(xd)

3×3 Matrix{Float32}:
 0.145793  0.939801  0.479926
 0.567772  0.577251  0.81655
 0.800538  0.38893   0.914135

6 Linear algebra

using BenchmarkTools, LinearAlgebra, Random

Random.seed!(257)

n = 2^14
# on CPU
x = rand(Float32, n, n)
y = rand(Float32, n, n)
z = zeros(Float32, n, n)
# on GPU
xd = MtlArray(x)
yd = MtlArray(y)
zd = MtlArray(z);

6.1 Dot product

# SP matrix dot product on GPU: tr(X'Y)
# why are there allocations?
bm_gpu = @benchmark Metal.@sync dot($xd, $yd)

BenchmarkTools.Trial: 693 samples with 1 evaluation.
 Range (min … max):  7.058 ms …  10.672 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     7.163 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   7.217 ms ± 280.146 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
   █▆▄▁                                                        
  ▅████▆▄▃▃▃▂▂▂▂▂▁▂▂▂▂▁▁▁▁▁▁▂▁▁▂▁▁▁▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂ ▂
  7.06 ms         Histogram: frequency by time        9.14 ms <
 Memory estimate: 26.30 KiB, allocs estimate: 956.

# SP matrix dot product on CPU: tr(X'Y)
bm_cpu = @benchmark dot($x, $y)

BenchmarkTools.Trial: 145 samples with 1 evaluation.
 Range (min … max):  34.256 ms …  35.923 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     34.705 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   34.707 ms ± 293.524 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
  ▃ ▂█          ▃▅▆ ▂ ▆▃▃  █▅       ▂                           
  ████▅█▅▄▁▄▁▄▇▄█████████▅▅██▄█▄▁▇▅▅█▅▄▄▇▁▄▁▁▅▁▁▁▁▄▁▁▁▁▁▁▁▁▁▁▅ ▄
  34.3 ms         Histogram: frequency by time         35.6 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

# speedup
median(bm_cpu.times) / median(bm_gpu.times)

4.844882645493413

6.2 Broadcast

# SP broadcast on GPU: z .= x .* y
# why is there allocation?
bm_gpu = @benchmark Metal.@sync $zd .= $xd .* $yd

BenchmarkTools.Trial: 553 samples with 1 evaluation.
 Range (min … max):  8.751 ms …  11.202 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     8.894 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   9.031 ms ± 286.161 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
   █▆▄▂▃▂                                                      
  ▆███████▅▄▄▃▂▃▁▃▂▂▄▄▅▅▆▅▅▅▆▇▅▄▅▃▃▄▃▂▁▂▁▃▁▁▁▁▁▁▁▁▁▁▁▁▂▃▂▁▂▂▂ ▃
  8.75 ms         Histogram: frequency by time        9.99 ms <
 Memory estimate: 5.34 KiB, allocs estimate: 197.

# SP broadcast on CPU: z .= x .* y
bm_cpu = @benchmark $z .= $x .* $y

BenchmarkTools.Trial: 141 samples with 1 evaluation.
 Range (min … max):  35.070 ms …  37.173 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     35.480 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   35.499 ms ± 240.034 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
             ▂▇█▂▇                                              
  ▃▄▃▃▃▃▇▃▄▆██████▆▆▄▄▁▃▁▃▃▃▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃ ▃
  35.1 ms         Histogram: frequency by time         36.9 ms <
 Memory estimate: 0 bytes, allocs estimate: 0.

# speedup
median(bm_cpu.times) / median(bm_gpu.times)

3.989440201602489

6.3 Matrix multiplication

# SP matrix multiplication on GPU
bm_gpu = @benchmark Metal.@sync mul!($zd, $xd, $yd)

BenchmarkTools.Trial: 6 samples with 1 evaluation.
 Range (min … max):  902.868 ms …   1.085 s  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     903.759 ms              ┊ GC (median):    0.00%
 Time  (mean ± σ):   939.934 ms ± 72.534 ms  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █                                                             
  █▁▁▁▁▁▁▁▁▁▁▁▆▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▆ ▁
  903 ms          Histogram: frequency by time          1.08 s <
 Memory estimate: 1.12 KiB, allocs estimate: 62.

For this problem size on this machine, we see GPU achieves a staggering 9 TFLOPS throughput with single precision!

# SP throughput on GPU
(2n^3) / (minimum(bm_gpu.times) / 1e9)

9.742394985320246e12

# SP matrix multiplication on CPU
bm_cpu = @benchmark mul!($z, $x, $y)

BenchmarkTools.Trial: 1 sample with 1 evaluation.
 Single result which took 12.845 s (0.00% GC) to evaluate,
 with a memory estimate of 0 bytes, over 0 allocations.

# SP throughput on CPU
(2n^3) / (minimum(bm_cpu.times) / 1e9)

6.847891934440028e11

We see >10x speedup by GPUs in this matrix multiplication example.

# cholesky on Gram matrix
# This one doesn't seem to work on Apple M2 chip yet
# xtxd = xd'xd + I
# @benchmark Metal.@sync cholesky($(xtxd))

# xtx = collect(xtxd)
# @benchmark cholesky($(Symmetric(xtx)))

GPU speedup of Cholesky seems unavailable at the moment.

7 Evaluation of elementary and special functions on GPU

# elementwise function on GPU arrays
fill!(yd, 1)
bm_gpu = @benchmark Metal.@sync $zd .= log.($yd .+ sin.($xd))
bm_gpu

BenchmarkTools.Trial: 508 samples with 1 evaluation.
 Range (min … max):  9.283 ms …  11.832 ms  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     9.779 ms               ┊ GC (median):    0.00%
 Time  (mean ± σ):   9.821 ms ± 392.345 μs  ┊ GC (mean ± σ):  0.00% ± 0.00%
     ▄▁         ▅▅▆██▅▇▅                                       
  ▄▇▇██▄▃▃▂▃▃▃▃▇████████▆▅▃▃▃▁▁▁▁▁▂▂▂▂▁▁▁▂▁▁▁▁▁▂▁▁▃▄▂▃▄▄▃▄▃▃▃ ▄
  9.28 ms         Histogram: frequency by time        10.9 ms <
 Memory estimate: 5.34 KiB, allocs estimate: 197.

# elementwise function on CPU arrays
x, y, z = collect(xd), collect(yd), collect(zd)
bm_cpu = @benchmark $z .= log.($y .+ sin.($x))
bm_cpu

BenchmarkTools.Trial: 3 samples with 1 evaluation.
 Range (min … max):  2.457 s …  2.471 s  ┊ GC (min … max): 0.00% … 0.00%
 Time  (median):     2.463 s             ┊ GC (median):    0.00%
 Time  (mean ± σ):   2.464 s ± 7.091 ms  ┊ GC (mean ± σ):  0.00% ± 0.00%
  █                    █                                 █  
  █▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁
  2.46 s        Histogram: frequency by time        2.47 s <
 Memory estimate: 0 bytes, allocs estimate: 0.

# Speed up
median(bm_cpu.times) / median(bm_gpu.times)

251.8307707741242

GPU brings great speedup (>50x) to the massive evaluation of elementary math functions.